Accurate Numerical Scheme Research Articles

Three-Dimensional Flow in a Driven Cavity Subjected to an External Magnetic Field

In this paper, we study the three-dimensional (3D) flow of an electrically conducting fluid in a cubic cavity with the top wall moving and subjected to an external magnetic field. The governing flow and electromagnetic field equations are integrated by a second-order space and time accurate numerical scheme, implemented on a graphics processing unit (GPU) with high parallel efficiency. Solutions for several Reynolds and Stuart numbers have been obtained on sufficiently fine grids to achieve grid independent solutions. As expected, the magnetic field significantly influences the circulation in the cavity and modifies the shape and locations of the primary and secondary eddies. The observed flow patterns are illustrated graphically as well as through selected line plots and tabulated data. With increasing magnetic field strength, the center of the primary eddy is seen to shift to the top right corner. Further, situations where the flow is unsteady in the absence of the magnetic field have become steady after a certain value of the magnetic interaction parameter.

Toward an Accurate Modeling of Hydrodynamic Effects on the Translational and Rotational Dynamics of Biomolecules in Many-Body Systems.

Proper treatment of hydrodynamic interactions is of importance in evaluation of rigid-body mobility tensors of biomolecules in Stokes flow and in simulations of their folding and solution conformation, as well as in simulations of the translational and rotational dynamics of either flexible or rigid molecules in biological systems at low Reynolds numbers. With macromolecules conveniently modeled in calculations or in dynamic simulations as ensembles of spherical frictional elements, various approximations to hydrodynamic interactions, such as the two-body, far-field Rotne-Prager approach, are commonly used, either without concern or as a compromise between the accuracy and the numerical complexity. Strikingly, even though the analytical Rotne-Prager approach fails to describe (both in the qualitative and quantitative sense) mobilities in the simplest system consisting of two spheres, when the distance between their surfaces is of the order of their size, it is commonly applied to model hydrodynamic effects in macromolecular systems. Here, we closely investigate hydrodynamic effects in two and three-body systems, consisting of bead-shell molecular models, using either the analytical Rotne-Prager approach, or an accurate numerical scheme that correctly accounts for the many-body character of hydrodynamic interactions and their short-range behavior. We analyze mobilities, and translational and rotational velocities of bodies resulting from direct forces acting on them. We show, that with the sufficient number of frictional elements in hydrodynamic models of interacting bodies, the far-field approximation is able to provide a description of hydrodynamic effects that is in a reasonable qualitative as well as quantitative agreement with the description resulting from the application of the virtually exact numerical scheme, even for small separations between bodies.

Exploration of the subcycle multiphoton ionization dynamics and transient electron density structures with Bohmian trajectories

An accurate three-dimensional numerical scheme for the De Broglie--Bohm framework of Bohmian mechanics is presented. This method is utilized to explore the subcycle multiphoton ionization dynamics of the hydrogen atom subject to intense near-infrared laser fields on the subfemtosecond time scale. The analysis of the time-dependent electron density reveals that several distinct density portions can be shaped and detached from the core within a half cycle of the laser field. As a complementary perspective, we identify several distinct groups of the Bohmian trajectories which represent the multiple detachments of the electron density at different times. The method presented provides very accurate electron densities and Bohmian trajectories that allow to uncover the origin of the formation of the transient and distinct electron structures seen in the multiphoton ionization processes.

Development of a mass conserving discretization technique for breakage problems and its convergence analysis

In this work an efficient and accurate numerical scheme based on the finite volume method has been introduced to approximate the pure breakage population balance equations. The scheme is designed to conserve the total mass of the particles in the system. The simplicity of both the discrete formulation and its coding are the key features of the new method. It is seen that besides conserving the total mass, the new scheme also gives a better prediction of the total number of the particles in the system as compared to the finite volume scheme of Kumar et al. (Appl Math Comput 219(10):5140–5151, 2013). Unlike [13], the scheme in this paper is computationally very efficient and robust to apply on both the uniform and nonuniform meshes. The development of the new scheme is completed by providing a detailed consistency and convergence analysis of the numerical solution. It is observed that the new scheme is of second order convergent independently of the type of meshes. Moreover, numerical results are compared against several test problems, which include the problems whose solutions are analytically tractable and those which are practically oriented. The mathematical results on convergence analysis of the new scheme has also been verified numerically.

Compact thermal modeling methodology for predicting skin temperature of passively cooled devices

Compact thermal modeling of microelectronic systems has recently attracted considerable attention. The present work aims at developing resistor-capacitor (RC) thermal models for predicting time-dependent surface temperature of a passively cooled device. The developed models mimic typical fanless systems cooled by dissipating heat to the surroundings through the enclosure back case by natural convection and radiation. In order to establish a baseline for checking the accuracy of the compact thermal model, the same problem was modeled using three-dimensional, transient, Navier–Stokes equations that were solved numerically by computational fluid dynamics (CFD) code. For constructing Foster RC-network ladders and to find best-fit thermal constants, temperature step responses of the hot-spot were obtained by applying known power pulses on the discrete heat source. Special attention was paid to the characteristics of the RC-network ladders for obtaining a reasonably accurate numerical scheme for real-time calculation of hot-spot surface temperature. In the present study it is demonstrated that the thermal constants (R, τ) are strong functions of the input power. An alternative usage of formulation for multi-ladder Foster RC-network, incorporating time-dependent thermal constants is show-cased. The suggested methodology can be used for non-linear problems involving time- and power-dependent boundary conditions.

Three dimensional HLL Riemann solver for conservation laws on structured meshes; Application to Euler and magnetohydrodynamic flows

In this paper we build on our prior work on multidimensional Riemann solvers by detailing the construction of a three-dimensional HLL Riemann solver. As with the two-dimensional Riemann solver, this is accomplished by introducing a constant resolved state between the states being considered, which introduces sufficient dissipation for systems of conservation laws. Closed form expressions for the resolved fluxes are provided to facilitate numerical implementation. This is accomplished by introducing a novel derivation of the multidimensional Riemann solver. The novelty consists of integrating Lagrangian fluxes across moving surfaces. This makes the problem easier to visualize in three dimensions. (A video introduction to multidimensional Riemann solvers is available on http://www.nd.edu/~dbalsara/Numerical-PDE-Course.)A robust and efficient second order accurate numerical scheme for three dimensional Euler and MHD flows is presented. The scheme is built on the current three-dimensional Riemann solver and has been implemented in the author's RIEMANN code. We demonstrate that schemes that are based on the three-dimensional Riemann solver permit multidimensional discontinuities to propagate more isotropically on resolution-starved meshes. The number of zones updated per second by this scheme on a modern processor is shown to be cost competitive with schemes that are based on a one-dimensional Riemann solver. However, the present scheme permits larger timesteps in three dimensions because of its inclusion of genuinely three-dimensional effects in the flow. For MHD problems it is not necessary to double the dissipation when evaluating the edge-centered electric fields.Accuracy analysis for three-dimensional Euler and MHD problems shows that the scheme meets its design accuracy. Several stringent test problems involving Euler and MHD flows are also presented and the scheme is shown to perform robustly on all of them. For the very first time, we present the formulation and solution of three-dimensional Riemann problems.

Accurate spectral numerical schemes for kinetic equations with energy diffusion

We examine the merits of using a family of polynomials that are orthogonal with respect to a non-classical weight function to discretize the speed variable in continuum kinetic calculations. We consider a model one-dimensional partial differential equation describing energy diffusion in velocity space due to Fokker–Planck collisions. This relatively simple case allows us to compare the results of the projected dynamics with an expensive but highly accurate spectral transform approach. It also allows us to integrate in time exactly, and to focus entirely on the effectiveness of the discretization of the speed variable. We show that for a fixed number of modes or grid points, the non-classical polynomials can be many orders of magnitude more accurate than classical Hermite polynomials or finite-difference solvers for kinetic equations in plasma physics. We provide a detailed analysis of the difference in behavior and accuracy of the two families of polynomials. For the non-classical polynomials, if the initial condition is not smooth at the origin when interpreted as a three-dimensional radial function, the exact solution leaves the polynomial subspace for a time, but returns (up to roundoff accuracy) to the same point evolved to by the projected dynamics in that time. By contrast, using classical polynomials, the exact solution differs significantly from the projected dynamics solution when it returns to the subspace. We also explore the connection between eigenfunctions of the projected evolution operator and (non-normalizable) eigenfunctions of the full evolution operator, as well as the effect of truncating the computational domain.

Exchange parameters of strongly correlated materials: Extraction from spin-polarized density functional theory plus dynamical mean-field theory

In this paper we present an accurate numerical scheme for extracting interatomic exchange parameters (${J}_{ij}$) of strongly correlated systems, based on first-principles full-potential electronic structure theory. The electronic structure is modeled with the help of a full-potential linear muffin-tin orbital method. The effects of strong electron correlations are considered within the charge self-consistent density functional theory plus dynamical mean-field theory. The exchange parameters are then extracted using the magnetic force theorem; hence all the calculations are performed within a single computational framework. The method allows us to investigate how the ${J}_{ij}$ parameters are affected by dynamical electron correlations. In addition to describing the formalism and details of the implementation, we also present magnetic properties of a few commonly discussed systems, characterized by different degrees of electron localization. In bcc Fe, treated as a moderately correlated metal, we found a minor renormalization of the ${J}_{ij}$ interactions once the dynamical correlations are introduced. However, generally, if the magnetic coupling has several competing contributions from different orbitals, the redistribution of the spectral weight and changes in the exchange splitting of these states can lead to a dramatic modification of the total interaction parameter. In NiO we found that both static and dynamical mean-field results provide an adequate description of the exchange interactions, which is somewhat surprising given the fact that these two methods result in quite different electronic structures. By employing the Hubbard-I approximation for the treatment of the $4f$ states in hcp Gd we reproduce the experimentally observed multiplet structure. The calculated exchange parameters result in being rather close to the ones obtained by treating the $4f$ electrons as noninteracting core states.

Accurate numerical schemes for approximating initial-boundary value problems for systems of conservation laws

Solutions of initial-boundary value problems for systems of conservation laws depend on the underlying viscous mechanism, namely different viscosity operators lead to different limit solutions. Standard numerical schemes for approximating conservation laws do not take into account this fact and converge to solutions that are not necessarily physically relevant. We design numerical schemes that incorporate explicit information about the underlying viscosity mechanism and approximate the physical-viscosity solution. Numerical experiments illustrating the robust performance of these schemes are presented.

Analysis of an open source quadtree grid shallow water flow solver for flood simulation

The shallow water flow solver within Gerris Flow Solver implements second-order accurate Gudunov type numerical schemes, with preserving the balance of source and flux terms on quadtree cut cell grids. The solver provides flexible grid generation, with using adaptive quadtree grids and the cut cell method, which can improve computational efficiency in 2D simulations. The solver, however, cannot implicitly discretize nonlinear friction without linearization since the current version of Gerris supports the implicit discretization only for linear friction. We compare implicit treatment for Manning's friction to the linearization. Simulation results using a benchmark test show the difference between two methods is only significant when the roughness coefficient is unrealistically high. Two real flood events, Malpasset dam break in France and Baeksan levee failure in Korea, are simulated using the quadtree grid based shallow water model, with adaptively refining meshes near water fronts and river boundaries. Comparison of simulation results with observations and measurements demonstrates that the grid adaptation can save approximately 85%–95% of the computational cost while preserving the accuracy. The research result shows that the linearized friction using the current version of Gerris can be applied for flood simulation and multiple inflow boundaries can be treated as source terms.

Review of multi-physics temporal coupling methods for analysis of nuclear reactors

The advanced numerical simulation of a realistic physical system typically involves multi-physics problem. For example, analysis of a LWR core involves the intricate simulation of neutron production and transport, heat transfer throughout the structures of the system and the flowing, possibly two-phase, coolant. Such analysis involves the dynamic coupling of multiple simulation codes, each one devoted to the solving of one of the coupled physics. Multiple temporal coupling methods exist, yet the accuracy of such coupling is generally driven by the least accurate numerical scheme. The goal of this paper is to review in detail the approaches and numerical methods that can be used for the multi-physics temporal coupling, including a comprehensive discussion of the issues associated with the temporal coupling, and define approaches that can be used to perform multi-physics analysis. The paper is not limited to any particular multi-physics process or situation, but is intended to provide a generic description of multi-physics temporal coupling schemes for any development stage of the individual (single-physics) tools and methods. This includes a wide spectrum of situation, where the individual (single-physics) solvers are based on pre-existing computation codes embedded as individual components, or a new development where the temporal coupling can be developed and implemented as a part of code development. The discussed coupling methods are demonstrated in the framework of LWR core analysis.

Tempered Fractional Sturm--Liouville EigenProblems

Continuum-time random walk is a general model for particle kinetics, which allows for incorporating waiting times and/or non-Gaussian jump distributions with divergent second moments to account for Lévy flights. Exponentially tempering the probability distribution of the waiting times and the anomalously large displacements results in tempered-stable Lévy processes with finite moments, where the fluid (continuous) limit leads to the tempered fractional diffusion equation. The development of fast and accurate numerical schemes for such nonlocal problems requires a new spectral theory and suitable choice of basis functions. In this study, we introduce two classes of regular and singular tempered fractional Sturm--Liouville problems of two kinds (TFSLP-I and TFSLP-II) of order $\nu \in (0,2)$. In the regular case, the corresponding tempered differential operators are associated with tempering functions $p_I(x) = \exp(2\tau) $ and $p_{II}(x) = \exp(-2\tau)$, $\tau \geq 0$, respectively, in the regular TFSLP-I and TFSLP-II, which do not vanish in $[-1,1]$. In contrast, the corresponding differential operators in the singular setting are associated with different forms of $p_I(x) = \exp( 2\tau) (1-x)^{1+\alpha} (1+x)^{1+\beta}$ and $p_{II}(x) = \exp( -2\tau) (1-x)^{1+\alpha} (1+x)^{1+\beta}$, vanishing at $x=\pm 1$ in the singular TFSLP-I and TFSLP-II, respectively. The aforementioned tempered fractional differential operators are both of tempered Riemann--Liouville and tempered Caputo type of fractional order $\mu = \nu/2 \in (0,1)$. We prove the well-posedness of the boundary-value problems and that the eigenvalues of the regular tempered problems are real-valued and the corresponding eigenfunctions are orthogonal. Next, we obtain the explicit eigensolutions to TFSLP-I and -II as nonpolynomial functions, which we define as tempered Jacobi poly-fractonomials. These eigenfunctions are orthogonal with respect to the weight function associated with TFSLP-I and -II. Finally, we introduce these eigenfunctions as new basis (and test) functions for spectrally accurate approximation of functions and tempered-fractional differential operators. To this end, we further develop a Petrov--Galerkin spectral method for solving tempered fractional ODEs, followed by the corresponding stability and convergence analysis, which validates the achieved spectral convergence in our simulations.

An energy-conserving second order numerical scheme for nonlinear hyperbolic equation with an exponential nonlinear term

We present a second order accurate numerical scheme for a nonlinear hyperbolic equation with an exponential nonlinear term. The solution to such an equation is proven to have a conservative nonlinear energy. Due to the special nature of the nonlinear term, the positivity is proven to be preserved under a periodic boundary condition for the solution. For the numerical scheme, a highly nonlinear fractional term is involved, for the theoretical justification of the energy stability. We propose a linear iteration algorithm to solve this nonlinear numerical scheme. A theoretical analysis shows a contraction mapping property of such a linear iteration under a trivial constraint for the time step. We also provide a detailed convergence analysis for the second order scheme, in the ℓ∞(0,T;ℓ∞) norm. Such an error estimate in the maximum norm can be obtained by performing a higher order consistency analysis using asymptotic expansions for the numerical solution. As a result, instead of the standard comparison between the exact and numerical solutions, an error estimate between the numerical solution and the constructed approximate solution yields an O(Δt3+h4) convergence in ℓ∞(0,T;ℓ2) norm, which leads to the necessary ℓ∞ error estimate using the inverse inequality, under a standard constraint Δt≤Ch. A numerical accuracy check is given and some numerical simulation results are also presented.

Phase-field approximations of the Willmore functional and flow

We discuss in this paper phase-field approximations of the Willmore functional and the associated \({\mathrm L}^{2}\)-flow. After recollecting known results on the approximation of the Willmore energy and its \({\mathrm L}^{1}\) relaxation, we derive the expression of the flows associated with various approximations, and we show their behavior by formal arguments based on matched asymptotic expansions. We introduce an accurate numerical scheme, whose local convergence is proved, to describe with more details the behavior of two flows, the classical and the flow associated with an approximation model due to Mugnai. We propose a series of numerical simulations in 2D and 3D to illustrate their behavior in both smooth and singular situations.

Analysis of stability, verification and chaos with the Kreiss–Yström equations

A system of two coupled PDEs originally proposed and studied by Kreiss and Yström (2002), which is dynamically similar to a one-dimensional two-fluid model of two-phase flow, is investigated here. It is demonstrated that in the limit of vanishing viscosity (i.e., neglecting second-order and higher derivatives), the system possesses complex eigenvalues and is therefore ill-posed. The regularized problem (i.e., including viscous second-order derivatives) retains the long-wavelength linear instability but with a cut-off wavelength, below which the system is linearly stable and dissipative. A second-order accurate numerical scheme, which is verified using the method of manufactured solutions, is used to simulate the system. For short to intermediate periods of time, numerical solutions compare favorably to those published previously by the original authors. However, the solutions at a later time are considerably different and have the properties of chaos. To quantify the chaos, the largest Lyapunov exponent is calculated and found to be approximately 0.38. Additionally, the correlation dimension of the attractor is assessed, resulting in a fractal dimension of 2.8 with an embedded dimension of approximately 6. Subsequently, the route to chaos is qualitatively explored with investigations of asymptotic stability, traveling-wave limit cycles and intermittency. Finally, the numerical solution, which is grid-dependent in space–time for long times, is demonstrated to be convergent using the time-averaged amplitude spectra.

Towards Eulerian Modeling of a Polydisperse Evaporating Spray Under Realistic Internal-Combustion-Engine Conditions

To assist the industrial engine design process, 3-D computational fluid dynamics simulations are widely used, bringing a comprehension of the underlying physics unattainable from experiments. However, the multiphase flow description involving a liquid fuel jet injected into the chamber is still in its early stages of development. There is a pressing need for a spray model that is time efficient and accurately describes the cloud of fuel and droplet dynamics downstream of the injector. Eulerian descriptions of the spray are well adapted to this highly unsteady configuration. The challenge is then to capture accurately both the evaporating spray polydispersity and its two-way mass, momentum and energy interactions with the surrounding gas phase in this framework. The Eulerian Multi-Size Moment (EMSM) model, a high-order (in size) moment model has proved to be well adapted for the treatment of polydispersity. Moreover, it requires less computational effort relative to competing methods, with a single section for the size phase space. Academic test cases have demonstrated its great potential for industrial applications using one-way coupling. Recent modeling and numerical development efforts have resulted in an extension to two-way coupling with an unconditionally stable and accurate numerical scheme, while preserving the size moment space. All these developments have been successfully verified through injection simulations in the context of laminar flow. In the present contribution, in preparation for comparisons with experimental data, an extension to two-way coupling to account for the droplet-gas turbulence interactions is developed and validated for academic cases using physical data for realistic internal combustion engine conditions.

Sound penetration into a hard-backed rigid porous layer: theory and experiments.

This study examines the sound field within a hard-backed rigid porous medium due to an airborne source. The total sound field can be approximated by two terms: A transmitted wave component arriving at the receiver directly through the porous interface, and a second transmitted wave component reflected from the rigid backing plane before reaching the receiver. These two components can be expressed in an integral form that is amenable to further analyses. A standard saddle path method is applied to evaluate the integral analytically, leading to a uniform asymptotic solution that allows the prediction of the sound field within the rigid porous medium. The validity of the asymptotic formula is verified by comparison with the numerical results computed by a more accurate wave-based numerical scheme. The asymptotic solution is shown to provide a convenient means of rapid and accurate computations of sound field within the rigid porous medium. The accuracy of the numerical solutions is further confirmed by comparison with indoor experimental results. The measurement data and theoretical predictions suggest that when the receiver is located near the bottom of the hard-backed layer, the reflection of the refracted wave gives rise to a significant contribution to the total sound field.

GPELab, a Matlab toolbox to solve Gross–Pitaevskii equations I: Computation of stationary solutions

This paper presents GPELab (Gross–Pitaevskii Equation Laboratory), an advanced easy-to-use and flexible Matlab toolbox for numerically simulating many complex physics situations related to Bose–Einstein condensation. The model equation that GPELab solves is the Gross–Pitaevskii equation. The aim of this first part is to present the physical problems and the robust and accurate numerical schemes that are implemented for computing stationary solutions, to show a few computational examples and to explain how the basic GPELab functions work. Problems that can be solved include: 1d, 2d and 3d situations, general potentials, large classes of local and nonlocal nonlinearities, multi-components problems, and fast rotating gases. The toolbox is developed in such a way that other physics applications that require the numerical solution of general Schrödinger-type equations can be considered. Program summaryProgram title: GPELabCatalogue identifier: AETU_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AETU_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 26552No. of bytes in distributed program, including test data, etc.: 611289Distribution format: tar.gzProgramming language: Matlab.Computer: PC, Mac.Operating system: Windows, Mac OS, Linux.Has the code been vectorized or parallelized?: YesRAM: 4000 MegabytesClassification: 2.7, 4.6, 7.7.Nature of problem:Computing stationary solutions for a class of systems (multi-components) of Gross–Pitaevskii equations in 1d, 2d and 3d. This program is particularly well designed for the computation of ground states of Bose–Einstein condensates as well as dynamics.Solution method:We use the imaginary-time method with a Semi-Implicit Backward Euler scheme, a pseudo-spectral approximation and a Krylov subspace method.Running time:From a few minutes for simple problems to a week for more complex situations on a medium computer.

MULTISCALE IMAGE ANALYSIS BASED ON ROBUST AND ADAPTIVE MORPHOLOGICAL SCALE-SPACES

Mathematical morphology is a powerful tool for image analysis; however, classical morphological operators suffer from lacks of robustness against noise, and also intrinsic image features are not accounted at all in the process. We propose in this work a new and different way to overcome such limits, by introducing both robustness and locally adaptability in morphological operators, which are now defined in a manner such that intrinsic image features are accounted. Dealing with partial differential equations (PDEs) for generalized Cauchy problems, we show that proposed PDEs are equivalent to impose robustness and adaptability of corresponding sup-inf operators, to structuring functions. Accurate numerical schemes are also provided to solve proposed PDEs, and experiments conducted for both synthetic and real images, show the efficiency and robustness of our approach.

Energy-conserving schemes for wind farm aerodynamics

Computational demands compel researchers to use coarse grids for the study of wind farm aerodynamics, which necessitates the use of accurate numerical schemes. Energy- conserving (EC) schemes are designed to enforce the conservation of Kinetic Energy (KE), an invariant property of incompressible flows. These schemes are numerically stable and free from artificial dissipation, even on coarse grids and could be used as an alternative to high-order pseudo-spectral schemes. This article details tests on EC schemes in the context of Large Eddy Simulations (LES). Results suggest that the accuracy of EC schemes is influenced by the Subgrid Scale model used for the LES. EC methods use central schemes that lead to dispersion, which is more apparent with a less-dissipative Scale Similarity model. Whereas, the purely dissipative Smagorinsky's model reduces the dispersion and generates a smoother solution but at the cost of accuracy in terms of predicting the KE. Although the impact of EC schemes and SGS models on the LES of wind farms is yet to be assessed, a simple LES of a model wind farm uncovers that EC schemes are quite suitable for wind farm aerodynamics.